How Fast is an Asteroid Traveling Toward Earth?

Click For Summary

Homework Help Overview

The discussion revolves around calculating the speed of an asteroid approaching Earth, with a focus on the necessary conditions for a rocket to intercept it effectively. The problem involves concepts from orbital mechanics and gravitational physics.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the application of gravitational equations to determine the asteroid's speed based on its distance from the sun. There are questions regarding unit consistency and the interpretation of results, particularly concerning the speed's realism.

Discussion Status

Some participants have provided guidance on unit conversion, emphasizing the importance of using SI units. Others suggest examining the asteroid's velocity at different distances from the sun to understand the range of possible speeds better. There is acknowledgment of the need to clarify the problem statement and its implications.

Contextual Notes

Participants note the ambiguity in the asteroid's distance from the sun and the potential impact on calculations. There is also mention of a misinterpretation in the problem statement regarding the required impact speed.

fleetingmoment
Messages
18
Reaction score
1

Homework Statement


A asteroid is hurtling towards Earth and humankind has decided to fire a nuclear warhead at it in order to avert disaster. In order be most effective the rocket carrying the warhead has to impact the asteroid at 40km/s. The rocket itself travels at 12km/s. What remains is to calculate the speed of the asteroid. During its elliptical orbit, the asteroid's greatest distance from the sun is 2.8 astronomical units (AU) and its smallest 1.00 AU. Its average distance from the sun is then 1.9 AU.

Homework Equations



1 x AU = 1.4960 * 1011m[/B]
The formula provided for calculating the speed of the asteroid is:
V2 = G * M * ((2/r) - (1/a))
where G *M = 1.327 * 1020 (gravitational constant times solar mass), r is the asteroid's distance from the sun (the book doesn't specify whether it is the greatest distance or the smallest) and a is its average distance from the sun.
2.8 * 1.4960 * 1011 = 4.1888 * 1011 = r
1.9 * 1.4960 * 1011 = 2.8424 * 1011 = a

The Attempt at a Solution


Plugging the relevant values into the equation thus:
V2 = 1.327 * 1020 * ((2/4.1888 * 1011) - (1/2.8424 * 1011))
gives 1.668 * 108
Taking the square root of both sides gives:
sqrt(V2) = sqrt(1.668 * 108) ⇔ V = 12915.1

Assuming my answer is correct, I've no idea what the given units are. Whether metres per second, or kilometres per hour, the value still seems incredibly high, given how fast asteroids actually travel. Have I gone wrong somewhere?
 
Last edited:
Physics news on Phys.org
Because G and M are SI units. All units must be SI units.
fleetingmoment said:
1 x AU = 1.4960 * 108km
km is not SI unit, you must change into meter
 
Hamal_Arietis said:
Because G and M are SI units. All units must be SI units.

km is not SI unit, you must change into meter
Thanks, Hamal_Arietis
The new value of 12915.1 seems a lot more realistic. Assuming it's also in metres per second. I'm going to divide by 1000 and conclude that the asteroid is traveling at 12.915 km/s.
 
If you find the asteroid velocity for both its nearest and furthest position from the Sun you will see that there is a range of velocities. I suppose you could find a particular distance where the rocket's speed and asteroid's speed combine to make the optimum collision speed.
 
  • Like
Likes   Reactions: Hamal_Arietis
gneill said:
If you find the asteroid velocity for both its nearest and furthest position from the Sun you will see that there is a range of velocities. I suppose you could find a particular distance where the rocket's speed and asteroid's speed combine to make the optimum collision speed.
Thanks, gneill,
I plugged in the value for 1 AU and got 36161.5 or 36.162 km/s. I was obviously hasty in concluding that the human race was doomed, based on the value for the farthest distance - especially since the question should have read 'impact the asteroid at at least 40km/s': something I missed when translating the question from the language I'm studying in.